Three kinds of particles on a single rationally parameterized worldline

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by  Vladimir V. Kassandrov

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Three kinds of particles on a single rationally parameterized worldline
Vladimir V. Kassandrova,1 and Nina V. Markovab,2
a Institute of Gravitation and Cosmology, Peoples’ Friendship University of Russia, Moscow, Russia
b Department of Applied Mathematics, Peoples’ Friendship University of Russia, Moscow, Russia
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We consider the light cone (‘retardation’) equation (LCE) of an inertially moving observer and a single
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worldline parameterized by arbitrary rational functions. Then a set of apparent copies, R- or C-particles,
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defined by the (real or complex conjugate) roots of the LCE will be detected by the observer. For any
a
J rational worldline the collective R-C dynamics is manifestly Lorentz-invariant and conservative; the latter
propertyfollowsdirectlyfromthe structureofVieta formulasfor the LCEroots. Inparticular,twoLorentz
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3 invariants, the square of total 4-momentum and total rest mass, are distinct and both integer-valued.
Asymptotically, at large values of the observer’s proper time, one distinguishes three types of the LCE

] roots and associated R-C particles, with specific locations and evolutions; each of three kinds of particles
h
p can assemble into compact large groups - clusters. Throughout the paper, we make no use of differential
- equations of motion, field equations, etc.: the collective R-C dynamics is purely algebraic
n
e
g
.
s PACS: 03.65.Fd, 11.30.-j,98.80.-k
c
i
s 1 Collective algebro-dynamics on generating set of equations, respectively. It is es-
y
h a single ‘polynomial’ worldline pecially interesting that all these laws follow solely
p from the structure of Vieta formulas for the whole
[
We intend to present some results in progress of system of roots, or from derivations of theseformu-

1 the ‘one electron Universe’ concept formulated by las w.r.t. the time parameter.
v
Stueckelberg [1], Wheeler and Feynman [2] and In the case of implicitly defined polynomial
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6 relatedtotheso-calleduniqueworldline. Thelatter worldline [3,4] theconsidered algebraic dynamics is
8 canbeeitherdefinedimplicitly,byasetofalgebraic Galilei-invariant and can be compared with New-
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equations containing the time parameter t [3,4], or ton’s collective N-point dynamics. On the con-
0
. in a familiar parametric way through consideration trary, in the second case of a polynomial worldline
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0 of the light cone equation (LCE) (equivalent to the implemented by the LCE of an inertially moving
7 well-known retardation equation) corresponding to observer one obtains a full set of Lorentz-covariant
1
an external observer [5]. In both cases, at some conservation laws for the total set of R-C parti-
:
v fixed value of t, one has a whole set of roots of the cles. Asymptotically, for rather great values of
i
X considered algebraic system which determine the the observer’s (proper) time T one encounters, in
r positions and, consequently, temporal dynamics of addition, the effects of ‘self-quantization’ of the
a
the collection of identical particlelike formations. admissible values of total rest mass and ‘clusteriza-
In the above cited papers it was shown that tion’ of particle-roots. The possible meaning of the
for an arbitrary worldline defined by polynomial obtained algebrodynamics 3 for realistic relativistic
functions (except a degenerate case of zero mea- physics requires further investigations.
sure) the arising collective dynamics of the system Adetailed exposition of theabovepresented re-
of particles-roots is necessarily conservative. This sults can also be found in [6]. Below we general-
means thata complete set of conservation laws (for izeourconsideration ofpolynomial dynamicstothe
total momentum, angular momentum and the ana- case of a worldline parameterized by arbitrary ra-
logue of total energy) holds for the system of two tional functions.
kinds (R- or C-) of particlelike formations repre- 3On theso-called algebrodynamical program see,e.g., [7–
sented by real and complex conjugate roots of the 9]
1e-mail: [e-mail]
2e-mail: [e-mail]
2
2 Lorentz-invariant algebraic dy-
Under Lorentz boosts, the observer’s (proper)
namics on a ‘rational’ worldline time T and the worldline parameter τ remain in-
variant,whilethecoordinatestransforminacanon-
In Euclidean 3D space E3, consider the LCE of an icalway. SincetheLCEremainsform-invariant, ef-
observer at rest, for simplicity, at the origin: fective collective kinematics (dynamics) of the RC-
ensemble is also Lorentz-invariant, and the results
(T −S(τ))2 −X(τ)2 −Y(τ)2−Z(τ)2 = 0, (1)
can be carried over to the reference frame of any
inertially moving observer. Note that all physical
where T represents the (proper) time of the ob-
server and S := X ,R~ := {X,Y,Z} = {X }, a = quantities shouldbehereconsideredasfunctionsof
0 a
theuniquetime–thepropertime T oftheobserver
1,2,3 are rational functions of the parameter τ:
(instead of distinct proper times of individual par-
(a)
S (τ) X (τ) ticles).
S(τ) := p+l , X (τ) := n+k . (2)
S (τ) a D (τ) Dispensing with the denominators in (1) and in
p n
the equation for the timelike coordinate x , x −
0 0
In numerators and denominators one has arbitrary
S(τ) = 0, and eliminating from these two the pa-
mutually irreducible polynomials in τ of corre-
rameter τ (say, making use of the corresponding
sponding degrees p,n,p + l,n + k, and it is as-
resultant structure), one arrives at a polynomial
sumed l > k ≥ 0 to obtain nondegenerate poly-
equation
nomial parts (and an adequate asymptotic behav-
P (x ,T) = 0 (4)
ior, see Section 3). Explicitly selecting polynomial N 0
and fractional parts for further consideration, one of degree N in x , in which the coefficients nec-
0
presents (2) in the form: essarily depend polynomially on T. The same pro-
X = f τl +g τl−1+h τl−2+···+ Sp−1, cedure also results in polynomial equations for any
0 0 0 0 of the three spacelike coordinates of particles-roots
S
p
{x }.
(a) a
X
X = f τk +g τk−1+h τk−2+···+ n−1.(3) Now, from the first two Vieta formulas (for the
a a a a
Dn sumsof roots x0(T),xa(T) and their squares)after
Under these assumptions, a finite set of N = necessary number of derivations w.r.t. T one ob-
2p+2n+2l (real and complex conjugate together) tains two Lorentz-invariant conservation relations
roots {τ (T)} of the LCE (1) is sought for, which of the form (µ = 0,1,2,3):
i
define the collective dynamics of N pointlike ob-
Xx˙µ = constant= Pµ,
jects associated with two types (R- or C, respec-
tively) of particles. Note that any pair of C- Xx˙µx˙µ+x¨µxµ = const = M, (5)
particles can be visualized in E3 according to the
where ‘dot’ designates ∂/∂T, and summing hence-
common real partsof conjugate roots andthusrep-
forth runs over all the i = 1,2,...,N roots of the
resents a composite (C-) particle of double mass.
generating LCE (1) 4. In the frame of reference
At particular instants of the observer’s time T
of the observer at rest only the component P is
0
some two of the roots become multiple and then nonzero, whereas P~ := {P } ≡ 0 (the center-of-
a
change their type (real to complex conjugate or
mass frame).
vice versa); the associated RC-particles merge and
In the case of a purely polynomial worldline [5]
undergo mutual transitions. Such ‘events’ can be
correspondingconservedquantities havebeeniden-
interpreted as the process of annihilation/creation
tifiedwiththe4-vector P oftotalenergy-momentum
µ
of a pair of R-particles (precisely, of a particle-
and the scalar of total rest mass (rest energy) M
antiparticle system) [1,3,5]. At the moments of
of the RC-system, respectively. In this case (for
merging, the effective twistor and electromagnetic
l > k)ithasbeenprovedthattheseinvariantsiden-
fields (the latter of Lienard-Wiehert type) become
tically satisfy the fundamental relativistic energy-
singularonanullstraightlineconnectingthepoints
momentum relation
of observation and merging. The situation resem-
bles the process of emission/adsorption of a light- PµPµ = M2 > 0 (6)
like carrier of interaction (a classical model of the 4For simplicity, in what follows the summing index i is
photon). not written out
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Moreover, ‘self-quantization’ of the total rest mass Finally, the other 3 components K~ := {K } =
a
values takes place for any polynomial worldline, M {M } oftherelativistictensoroftotalangularmo-
oa
being always positive, integer and equal to the full mentum M := (x x˙ − x x˙ ) also preserve
[µν] P µ ν ν µ
number of particles-roots, M = 2l. their values in time, according to numerous com-
In the case of rational worldlines in question, putational tests. Unfortunately, neither the proof
the ‘self-quantization’ effect still takes place. How- of this fact nor the analytical formulas relating val-
ever,inthiscaseitturnsoutthat P = 2p+2l, P~ = ues of K~ with the coefficients of parameterizing
0
0, and for the square of 4-momentum one gets functions (3) have not yet been obtained.
P2 = P Pµ = (2p+2l)2 > 0, (7)
µ
3 Rationally parameterized world-
whereas for the second invariant one obtains line: three kinds of RC-particles
M =2p+2l−2n. (8)
Asymptotically, at large values of the observer’s
so that the inequality proper time T, three classes of roots can be distin-
guished,eachdefiningitsownsetofRC-particlelike
P2 > M2 (9) formations. Specifically, 2l roots (P-roots) are in-
herited from the structure of polynomial parts of
holds for any R-C dynamical system.
generating functions and reproduce, on the whole,
The above presented results can be proved by
the asymptotic properties of the corresponding
analyzing the structure of resultants, that is, de-
roots-particles for the case of a purely polynomial
terminants of corresponding Sylvester matrices.
worldline [5]. They are defined by the solutions for
Quite a similar procedure has been used and ex-
which one has |τ| ∼ T >> 1, and the correspond-
posed in details in [5] for the case of purely polyno-
ing particles asymptotically join into pairs which
mial worldlines. Note that the second invariant M
then assemble into large groups – clusters (under
(identified previously with total rest mass) can be
thecondition of mutually multiple l and k), under-
negative: this might be related to the contribution
goingmutualrecessionwithretardation(fordetails
(∼ 2n) of negative interaction energy, a possible
see [5,6]).
relativistic analogue of the potential energy.
In the case of rational worldlines in question,
As for the values of total angular momentum
there exists, however, another class (S) of pairs of
M~ , its conservation is affirmed by numerous com-
roots-particles corresponding to rather small val-
putationalexperimentsforworldlineswithdifferent
ues |τ| ∼ 1. The constituent particles asymptoti-
p,n,l > k and widely varying coefficients. More-
cally approach one another and one of the p zeros
over, for some specific combinations of degrees one
of the polynomial denominator S (τ) of the time-
p
manages to establish phenomenological formulas
like coordinate function S(τ). There are exactly
for its values. For example, in the case l = 2k−1
2p roots-particles of this kind: they are located
(and arbitrary p > 1,n > 1) one obtains
at small distances |x | ∼ 1 and move with rather
a
2 small velocities v ∼ 1/T2 << 1. Finally, the third
M~ =− [ f~×~g ], (10)
kind (D) of roots-particles (2n in number) relates
f
0
to the zeros of the ‘spatial’ polynomial denomina-
while for l = 2k−2,
tor D (τ). The arising pairs are located at great
n
distances andmovewithultra-relativistic velocities
2g 4
M~ = 0[ f~×~g ]− [ f~×~h ], (11)
v ∼ 1.
f2 f
0 0
Itisespeciallyinterestingwhenthepolynomials
where f~:= {fa},~g := {ga},~h := {ha} are3-vectors - denominators Sp(τ) and/or Dn(τ) possess mul-
composed from the correspondingolder coefficients tiple roots (of possibly great multiplicity). Then,
in expressions for the parameterizing functions (3). apart from the clusters related to P-class particles,
Note that the purely fractional parts of these func- oneencounterstwoothersortsofclusterswhichare
tions do not contribute to the angular moment val- due to the asymptotic generation of roots close to
ues at all. For any l ≥ 2k, the total angular mo- those of denominators Sp(τ) and/or Dn(τ). Parti-
mentum is also zero. clesofthefirst(S)classcomposebiggroupslocated
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near the observer andpossessingnonrelativistic ve- Using a similar procedure for spatial coordinates,
locities. Clusters of he second (D) class consist of we get at the end
distant particles moving with velocities asymptot- 209828990102272
ically approaching the speed of light (contrary to Pa =0, Wa := Xx˙2a+xax¨a = {852207569447017,
P-clusters for which velocities asymptotically tend 231793932340096 581026160921836
, }, (17)
to zero). More details of the general situation will 852207569447017 852207569447017
be illustrated on a typical example presented be-
so that for Lorentz invariants (7),(8) one has
low.
P2 = P Pµ = 1936 = 442 ≡ (2p+2l)2, M =
µ
W −W −W −W = 32 ≡ 2p+2l−2n. (18)
4 Example and discussion 0 1 2 3
Now, for the total angular momentum one obtains
As anexampleof arationally parameterized world-
M~ = {−474,−192,−286}, (19)
line, let us choose, in a random way, polynomial
parts with (mutually multiple) l = 16,k = 9 and
infullagreementwithEq.(11). Finally,theremain-
supplementthese by fractional parts with, say, p = ing components K~ of the angular momentum ten-
6,n = 6. Besides, let the denominator Sp(τ) pos- sor are rational numbers of order 107.
sess a root τ = 2 of multiplicity 4 and τ = 5 of
At large values of the observer’s time, say, at
multiplicity 2, while the denominator Dn(τ) has a T = 1014, there are 7 simple 5 groups - clusters
root τ = −1 of multiplicity 3 and τ = −7 of mul-
consisting of 2 composite C-particles (7× 4 = 28
tiplicity 3. Specifically, let us take the coefficients
roots) and 2 pairs of R-particles (2×2 = 4 roots)
of parameterizing functions (3) as
corresponding altogether to 2l = 32 roots of P-
f = 1, g = −3, h = −4; f~= {1,2,−3}, class for which |τ| ∼ T >> 1, and the LCE can be
0 0 0
approximated by the reduced form (T −S(τ))2 ∼
~g = {−11,17,13}, ~h= {−3,7,−9}; (12)
0. One particle in any pair corresponds to re-
tarded (T−ℜ(S)> 0) while the other to advanced
and, in the numerators of the fractional parts,
(T −ℜ(S)< 0) solutions to the LCE. All these are
Sp−1 = −4τ5+..., Xn−1 = −11τ5+..., located at a distance R ∼ Tk/l = T9/16 (see [5])
Yn−1 = 5τ5+..., Zn−1 = 14τ5+..., (13) from the observer and slowly recess, whereas mu-
tual distances slowly decrease (as compared with
while the denominators are of the form
R). For more details see [5,6].
Further, one has 8 roots of S-class forming the
S = (τ−2)4(τ−5)2, D = (τ+1)3(τ+9)3. (14)
p n cluster of two R-pairs (4 roots) and one C-pair (4
roots) of particles, and 4 roots of S-class consisting
(the omitted terms are of no need for the reader).
of a pair of C-particles, 2p = 12 of S-roots alto-
Reducingthel.h.s. oftheLCEandtheequation
gether. They correspond to the values of τ very
x −S(τ) = 0 to common denominators and calcu-
0
close to the zeros (2 and 5) of the denominator
lating the resultant of the two polynomials repre-
S . Both clusters are rather close to the observer
sented by thecorrespondingnumerators, we obtain p
(R ∼ 104 or 106, respectively) as well as the con-
the equation
stituent RC-particles which move with negligibly
Ax56+BTx55+CT2x54+··· = 0, (15) small velocities (v ∼ 10−13 and 10−7). Rather a
0 0 0
large number of roots-particles (8) in the first clus-
(A,B,C are very large integers whose values are ter correlates with the multiplicity 4 of the root
not important here), from which we get the whole τ = 2 of the denominator S . One half of the
p
number of particles-roots N = 56 = 2p+2l +2n particles-roots corresponds again to retarded and
and,makinguseofVietaformulas,theconservation the other half to advanced solutions.
laws for the derivatives of the timelike coordinate Finally, there are two groups of 6 roots of D-
x˙ := dS/dT, class corresponding to zeros −9 and −1 of the de-
0
nominator D . However, due to the odd degree
n
P0 = −B/A= 44, W0 := Xx˙20+x0x¨0 = 5Multiparticle clusters require a great multiplicity of l
(B/A)2−2C/A = 44 = P0 ≡ 2p+2l. (16) and k; corresponding situation is illustrated in [5,6]
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of multiplicity of both zeros, each group of roots treated as a sort of a ‘toy model’. On the other
gives rise to 2 clusters of 3 particles disposedat op- hand, the fact that all the above presented proper-
posite directions of the celestial sphere. Thetypical tiesfollow fromonlypurelymathematical consider-
distances of all clusters are R ∼ T, and velocities ations, free of any phenomenological speculations,
of all particles are very close to the speed of light, seems quite remarkable and unexpected. The ex-
v ∼ 1. Particles in the first two clusters corre- istence of such structures cannot be ignored: they
spond to advanced solutions of the LCE; they all certainly deserve further mathematical investiga-
approach the observer (from opposite directions). tions and proper physical interpretations.
Conversely, particles in other two clusters are re-
Theauthors aregrateful to Joseph A.Rizcallah
tarded and recess from the observer (in opposite
for fruitful discussions and comments.
directions): thesestructuresbearmostresemblance
of ‘cosmological’ objects.
References
To conclude, we have discovered some remark-
able properties of algebraic structures related to [1] E.C.G. Stueckelberg // Helv. Phys. Acta, 14, 588
the specific form of the LCE and rationally pa- (1941); 15, 23 (1942)
rameterized worldlines. Thesepropertieswidenthe [2] R.P. Feynman // Science, 153, 699 (1966)
previously obtained [5] possibilities for description R.P. Feynman // Phys. Rev., 76, 749 (1949)
of N-point collective relativistic dynamics on the
[3] V.V. Kassandrov and I.Sh. Khasanov // J.
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the polynomial case, the algebraic dynamics on a
arXiv:1211.7002
rational worldline (asymptotically) involves 3 R-
[4] V.V. Kassandrov, I.Sh. Khasanov and N.V.
and 3 C- distinct kinds of particlelike formations,
Markova // Bulletin of Peoples’ Friend.
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distantobjects movingwithultra-relativistic veloc-
Markova // J. Phys. A: Math. Theor., 48, 395204
ities and slowly moving nearby objects.
(2015); arXiv:1501.01606
Ingeneral,allthesearejoinedintolargeclusters
[6] I.Sh. Khasanov PhD Thesis // Moscow: Peoples’
consisting of a lot of constituent particles-roots.
Friend. Univ. of Russia. 2015 (in Russian). Avail-
Contrary to the ‘polynomial’ case (in which re-
able at www.dissovet.rudn.ru/web-local.
cession takes place along one or two privileged 3-
[7] V.V. Kassandrov Algebraic Structure of Space-
directions [5]), particles of S- and D-classes can
Time and Algebrodynamics // Moscow: Peoples’
at late times be distributed isotropically. As a
Friendship University Press, 1992 (in Russian)
whole, the pattern of evolution partly resembles
Milne’s kinematic relativity [10]; comparison of the [8] V.V. Kassandrov // Gravit. & Cosmol., 3, 216
(1995); gr-qc/0007027
two models will be considered elsewhere.
However, of utmost interest from a physical [9] V.V. Kassandrov // Phys. Atom. Nuclei, 72, 813
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conservative nature of the collective RC-dynamics
[10] E.A. Milne Relativity, Gravitation And World-
inherited from the polynomial case, as well as the Structure// Oxford: Clarendon Press, 1935.
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Lorentz invariants. Remarkably, here the square of tivity, Gravitation and World-Structure//Oxford:
Clarendon Press, 1948
4-momentum is always greater than the total rest
mass (see (9)), so that the famous mass-shell con-
dition in SR (valid, in fact, only for free or locally
colliding particles) is generalized here to the case
of an arbitrary interacting N-particle system.
Of course, one can hardly believe that the
elaborated construction has an explicit relation to
the real physical world: more probably, it can be